Page:Carroll - Euclid and His Modern Rivals.djvu/189

. § 3.]

P. 12. Th. 8. 'The angles of intersection of a Transversal with two sepcodal Directives are equal'.

Min. Do you prove that by Mr. Wilson's method?

Nie. Not quite. He does it by transferring an angle: we do it by divergence of directions.

Min. I prefer your method. All it needs to make it complete is the proof of the reality of such Lines: but that is unattainable, and its absence is fatal to the whole system. Nay, more: the fact, that the reality of such Lines leads by a logical necessity to the reality of Lines which make equal angles with any transversal, reacts upon that unfortunate Axiom, and destroys the little hope it ever had of being granted without proof. In point of fact, in asking to have the Axiom granted, you were virtually asking to have this other reality granted as axiomatic—but all this I have already explained (p. 125).

P. 13. Th. 10. 'If a Transversal cut two Directives and make the angles of intersection with them equal, the Directives are sepcodal'.

Min. The subject of your Proposition is indisputably real. If then you can prove this Theorem, you will thereby prove the reality of sepcodal Lines. But I fear you have assumed it already in Th. 7. There is still, however, a gleam of hope: perhaps you do not need Th. 7 in proving this?